Sublogarithmic-transexponential series

TL;DR

This paper constructs a new ordered differential field of sublogarithmic-transexponential series, extending existing frameworks to include a transexponential function and its derivatives, and demonstrates its properties and closure under composition.

Contribution

It adapts the van den Dries, Macintyre, and Marker construction to build a new ordered differential field with a transexponential function, expanding the theory of transseries.

Findings

Built an ordered differential field of sublogarithmic-transexponential series.

Proved germs of $ ext{transexp}$-terms are ordered at infinity.

Established closure under composition for the constructed Hardy field.

Abstract

We adapt the construction of the field of logarithmic-exponential transseries of van den Dries, Macintyre, and Marker to build an ordered differential field of sublogarithmic-transexponential series. We use this structure to build a transexponential Hardy field closed under composition. Specifically, we prove that the germs at $+\infty$ of $\mathcal{L}_{\mathrm{transexp}}$-terms in a single variable are ordered, where $\mathcal{L}_{\mathrm{transexp}}$ is a language containing $\mathcal{L}_{\mathrm{an}}(\exp,\log)$ with new symbols for a transexponential function, its derivatives, and their compositional inverses.